1. Field of the Invention
The present invention relates to a nonlinear function generation apparatus for generating a nonlinear function having hysteresis characteristics and a musical tone synthesis apparatus utilizing the same and, more particularly, to a musical tone synthesis apparatus using a so-called delayed feedback type decay tone algorithm used in, e.g., an electronic musical instrument and a nonlinear function generation apparatus suitable therefor.
2. Description of the Prior Art
Conventionally, as a so-called digital sound source used in, e.g., an electronic musical instrument, a musical tone synthesis apparatus using a so-called delayed feedback type decay tone synthesis algorithm for synthesizing musical tones by introducing data such as initial waveform data, impulse signal data, nonlinear signal data, and the like into a closed loop including a delay circuit, and performing feedback arithmetic processing of the introduced data is known (e.g., Japanese Patent Laid-open Sho. No. 63-40199).
The musical tone synthesis apparatus physically approximates a mechanical vibration system of an acoustic musical instrument such as a pipe of a wind instrument, strings of a stringed instrument, or the like by an electrical circuit. The apparatus receives a nonlinear signal corresponding to a reed or embouchure (mouthpiece) of a wind instrument or a movement of a contact between a bow and a string of a bowed instrument, thereby naturally and faithfully synthesizing a tone of the wind instrument or the bowed instrument as well as a change in tone magnitude.
FIG. 10 shows the relationship between an external force F given to a string by a bow of a bowed instrument, and a displacement velocity V given to the string by the external force F. When the external force F is near zero, since contribution of a static friction is dominant, the displacement velocity V is proportional to the external force F. When a given external force or more is applied, a dynamic friction becomes dominant, and the displacement velocity V is rendered constant or is inverse proportional to the external force F. Upon transition from the static friction to the dynamic friction, since a degree of contribution of the external force F to the displacement velocity V of a string is abruptly changed, external force F vs. displacement velocity V characteristics are represented by a nonlinear curve, as shown in FIG. 10. When the external force F is decreased, the displacement velocity V changes nonlinearly upon transition from the dynamic friction to the static friction. In this manner, in a synthesis algorithm of a bowed instrument, a nonlinear signal having a hysteresis characteristic shown in FIG. 10 is required. It is preferable to finely control this nonlinear signal according to a bow pressure or a bow velocity.
The present inventor has previously proposed, as a signal source of such a nonlinear signal having a hysteresis, a system wherein a characteristic function based on the static friction and that based on the dynamic friction are switched according to an input value of, e.g., an external force, and a threshold level for switching the functions is shifted according to a change direction of the input value (Japanese Patent Application Hei. No. 1-192708: U.S. Pat. application Ser. No. 07/557,963). However, hardware or software for realizing this system is complicated.
The present inventor has also previously proposed a system wherein a nonlinear signal generation apparatus is inserted in a feedback loop to provide a hysteresis to an output signal (Japanese Patent Application Hei. No. 1-194544: U.S. Pat. application Ser. No. 07/557,963). This system suffers from another demerit although it has a simple arrangement.
FIG. 11 shows a nonlinear signal generation apparatus for a bowed string algorithm which generates a hysteresis by a feedback loop. In FIG. 11, a nonlinear table 71 generates a nonlinear function, as shown in FIG. 12. This nonlinear function is defined by an almost straight line portion having a negative inclination .alpha. near an origin, and curve portions having a positive inclination at two ends of the straight line portion. A feedback circuit 72 shown in FIG. 11 has a positive gain .beta..
An I/O transfer function of such a nonlinear system will be examined below in units of portions having positive and negative inclinations of the nonlinear functions, respectively. In a portion having the positive inclination, a hysteresis is generated by the positive feedback gain .beta.. In a portion having the negative inclination .alpha., since a feedback loop serves as a negative feedback (NFB) loop, a total gain (transfer function) G.sub.NFB is given by: ##EQU1##
In the bowed string algorithm, since the gain of the straight line portion of the nonlinear function must be a constant (normally, about -1 or -2), if G.sub.NFB given by the above equation is represented by, e.g., -1, we have: ##EQU2## In consideration of stability of the system under the condition in that .alpha. is positive and .beta. is negative, if a loop gain .alpha..beta. is too large, the NFB system often causes a parasitic oscillation. As a result, .alpha. is settled to be about -1.1, and .beta. is settled to be about 0.09. However, when a positive feedback (PFB) system is constituted by portions having positive .alpha. at two ends of the straight portion of the nonlinear function, the above-mentioned feedback amount cannot provide a sufficient PFB amount, and a hysteresis cannot be satisfactorily generated. In other words, with the arrangement shown in FIG. 11, a feedback amount of the PFB system must be increased to obtain a sufficient hysteresis, and must be decreased to attain stable NFB. However, it is almost impossible to attain both generation of a sufficient hysteresis and stability of the system.